This article is concerned with the existence of solution of a certain non-autonomous linear delayed-advanced differential equation. The main objective is to provide the proof of a theorem introduced in Lima et al. (J. Comput. Appl. Math. 234(9):2732–2744, 2010) about existence of solution of a class of mixed-type functional differential equations (MTFDEs). It is an effort to complete the theoretical basis of some computational methods introduced earlier to solve numerically such equations, which were deduced making use of that theorem.